Lecture 4 Section – Tue, Jan 23, 2007

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Lecture 4 Section 1.4.1 – 1.4.2 Tue, Jan 23, 2007 What’s in the Bag? Lecture 4 Section 1.4.1 – 1.4.2 Tue, Jan 23, 2007

Two Bags -1000 10 20 30 40 60 1000 50 Bag A Bag B

The Hypotheses H0: The shown bag is Bag A. H1: The shown bag is Bag B. -1000 10 20 30 40 60 1000 50 Bag A Bag B

Decisions and Errors It is Bag A It is Bag B Win $1890 Lose $560 (Type II) We decide it is Bag A Lose $560 (Type I) Win $1890 We decide it is Bag B

Decision Rules Decision Rule

Decision Rules In our DieA/DieB example with one roll, our decision rule was to accept H0 if the die rolled a 1. For that rule,  = 0.20 and = 0.10. With two rolls, our decision rule was to accept H0 if the average was 1 or 3.5. For that rule,  = 0.04 and = 0.19.

Possible Decision Rules Reject H0 if the voucher is worth either $60 or $1000, i.e.,  $60. What is ? What is ? -1000 10 20 30 40 60 1000 50 Bag A Bag B

Rejection and Accepance Regions Rejection Region Acceptance Region Critical Value The critical value itself will be included in the rejection region.

Decision Rule #1 $60

Decision Rule #1 $60 Critical Value

Decision Rule #1 $60 Rejection Region Critical Value

Decision Rule #1 $60 Rejection Region Acceptance Region Critical Value

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute   = 1/20 = 0.05 Bag A Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A -1000 10 20 30 40 50 60 1000 Bag B -1000 10 20 30 40 50 60 1000

Compute  Bag A  = 12/20 = 0.60 Bag B -1000 10 20 30 40 50 60 1000

Extreme Values Direction of Extreme Most Extreme Value Next-most Extreme Value, etc.

Another Decision Rule Decision Rule #2 Reject H0 if the voucher is worth  $50. What is ? What is ? -1000 10 20 30 40 60 1000 50 Bag A Bag B

Another Decision Rule Decision Rule #2 Reject H0 if the voucher is worth  $50. What is ? What is ? -1000 10 20 30 40 60 1000 50 Bag A Bag B

 vs.  If we decrease , we will increase , and Is it possible to decrease both  and  at the same time?

Two-Sided Rejection Regions See Example 1.6, p. 26. Bag E Bag F 1 2 3 4 5 6 7 8 9 10